Well, let us look at a couple of examples:
How about 3/4 divided by 1/8...
that really means, "how many groups of 1/8 can we get from 3/4", right? So,
if we think about what an eighth is, we can remember that there are two of
them in a fourth, and we have three fourths, so we have 2 X 3 eighths, or 6/8.
Now, if you flip the 1/8 you get eight wholes... 3/4 times 8 is 24/4, or 6.
What really happened when you flipped that second number?
Suppose you multiplied both fractions by 8 (that would be like multiplying by
8/8, since it is a division problem -- do you see why?)
So now you have (3/4)8 divided by (1/8)8. That would not change the quotient,
because you are really multiplying by 8/8, and that is the same as
multiplying by 1.
That gives you (3/4)8 divided by 8/8,
which is the same as (3/4)8 divided by 1, which is the same as (3/4)8, which
is what we started with when we flipped the second fraction and multiplied.
I think if we are teaching students about division by fractions it would be
most important to make sure they understand that division by a fraction is
the same as division by a whole number. We want the second fraction to be
worth "one" so we multiply by its reciprocal. Explaining that part is very
important. When we teach students algorithms that seem magical, we do not
give students an opportunity to use their number sense.
-Gail, for the Teacher2Teacher service